In the study of spectral theory, an interesting question to ask is how complex the spectrum is. If we consider the topological structure of a compact set, there are two extremes, one is the interval, one is the Cantor set. Interval is simple, intuitive, while the Cantor set is complex, counterintuitive. For \( b=1 \) , it is known that if the frequency \( \alpha \in {\mathbb {Q}}^d \) , the spectrum of \( H_{v,\alpha ,\theta } \)  is a disjoint union of at most finite compact intervals. Furthermore, Bethe–Sommerfeld Conjecture states that for \( b\ge 2 \) , the spectrum of the \( b \) -dimensional Schrödinger operator with periodic potential contains only finitely many gaps, partial advances see [1]}, [2]}. But for quasi-periodic case, the spectrum contains only finitely many gaps is a novel phenomenon. For \( b=1 \) , Goldstein-Schlag-Voda [3]} showed that for multi-frequency case, the spectrum can be a single interval. While for \( b\ge 2 \) , things become easier. It is known that the spectrum of separable \( b \) -dimensional Schrödinger operators are sums of the spectrum of single 1-dimensional operators (consult section REF ). Thus if we assume the potential \( v \)  of \( H_{v,\alpha ,\theta } \)  is separable potentials, intuitively, even though the spectrum of 1-dimensional operators is a Cantor set, if those gaps in every Cantor set have good control, then their sums can still produce an interval, thus it is natural to expect a condition such that the interval spectrum occurs.
The famous Newhouse Gap Lemma [4]} provides a tool to link the gap information for a single Cantor set and their sums, thus our gap estimates can be applied to show the interval spectrum. This approach has been studied by Takase [5]}, who showed that if any 1-dimensional Schrödinger operator with analytic potential frequency-dependent close to constants, then their sums is an interval. We extend this work to infinite differentiable regime, that is, the following result:
